Theorem ontric3g 44110

Description: For all 𝑥, 𝑦 ∈ On, one and only one of the following hold: 𝑥 ∈ 𝑦, 𝑦 = 𝑥, or 𝑦 ∈ 𝑥. This is a transparent strict trichotomy. (Contributed by RP, 27-Sep-2023.)

Assertion

Ref	Expression
ontric3g	⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑥 ∈ 𝑦 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 = 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 ∈ 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥)))

Distinct variable group:   𝑥,𝑦

Proof of Theorem ontric3g

Step	Hyp	Ref	Expression
1		orcom 883	. . . . . . 7 ⊢ ((𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥) ↔ (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥))
2	1	a1i 11	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥) ↔ (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥)))
3		onsseleq 6391	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ⊆ 𝑥 ↔ (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥)))
4		ontri1 6384	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
5	2, 3, 4	3bitr2d 310	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥) ↔ ¬ 𝑥 ∈ 𝑦))
6	5	con2bid 357	. . . 4 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑥 ∈ 𝑦 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥)))
7	6	ancoms 463	. . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ∈ 𝑦 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥)))
8	4	ancoms 463	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
9		ontri1 6384	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝑥))
10	8, 9	anbi12d 643	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑦) ↔ (¬ 𝑥 ∈ 𝑦 ∧ ¬ 𝑦 ∈ 𝑥)))
11		eqss 3954	. . . 4 ⊢ (𝑦 = 𝑥 ↔ (𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑦))
12		ioran 999	. . . 4 ⊢ (¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (¬ 𝑥 ∈ 𝑦 ∧ ¬ 𝑦 ∈ 𝑥))
13	10, 11, 12	3bitr4g 317	. . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦 = 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥)))
14		equcom 2041	. . . . . . 7 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
15	14	orbi2i 925	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦))
16	15	a1i 11	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦)))
17		onsseleq 6391	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ⊆ 𝑦 ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦)))
18	16, 17, 9	3bitr2d 310	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥) ↔ ¬ 𝑦 ∈ 𝑥))
19	18	con2bid 357	. . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦 ∈ 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥)))
20	7, 13, 19	3jca 1144	. 2 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ 𝑦 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 = 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 ∈ 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥))))
21	20	rgen2 3205	1 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑥 ∈ 𝑦 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 = 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 ∈ 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥)))

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   ∈ wcel 2145  ∀wral 3079   ⊆ wss 3907  Oncon0 6350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-tr 5213  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-ord 6353  df-on 6354

This theorem is referenced by: (None)